Suppose C is a category with a symmetric monoidal structure, which we will refer to as the smash product. Then the Picard category is the full subcategory of objects which have an inverse under the smash product in C, and the Picard group Pic(C) is the collection of isomorphism classes of such invertible objects. The Picard group need not be a set in general, but if it is then it is an abelian ...

Smooth toric Fano varieties are classified up to dimension 4. In dimension 2, there are five toric Del Pezzo surfaces: P, P1×P1, and Si, the blowup of P in i points, for i = 1, 2, 3. There are 18 toric Fano 3-folds [2, 20] and 124 toric Fano 4-folds [4, 17]. In higher dimensions, little is known about them and many properties that hold in low dimensions are not known to hold in general. Let X b...

In this paper, we consider fractional differential equations with the new derivative involving a nonsingular kernel, namely, Caputo-Fabrizio derivative. Using successive approximation method, prove an extension of Picard-Lindelöf existence and uniqueness theorem for derivative, which gives set conditions, under initial value problem has unique solution.

Rational Curves on Fano Threefolds of Picard Number One

We prove a geometric logarithmic derivative lemma for rigid analytic mappings to algebraic varieties in characteristic zero. We use the lemma to give a new and simpler proof (at least in characteristic zero) of Berkovich’s little Picard theorem [Ber, Theorem 4.5.1], which says there are no nonconstant rigid analytic maps from the affine line to non-singular projective curves of positive genus, ...